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POLARIZED  LIGHT  FROM  DIFFERENT  SOURCES.            253
supposed not to lie beyond the limits 90 and +90, we get from the first equation /3' =  /3, or /3' =  90  /3, + or  according as /3 is positive or negative. Now it is plain that any one solution must be expressed analytically in two ways, in which the values of /3' are complementary, and the values of a differ by 90, since either principal axis of the ellipse belonging to the second stream may be that whose azimuth is a'. Accordingly, we may reject the second solution as being nothing more than the first expressed in a different way, and may therefore suppose /3' =  @. The second and third equations then give cos 2a' =  cos 2a, sin 2a =  sin 2a, and therefore a and a! differ by 90. The equations are indeed satisfied by /3 = - f =  45, but this solution is only a particular case of the former.
It follows therefore that common light is equivalent to any two independent oppositely polarized streams of half the intensity ; and no two independent polarized streams can together be equivalent to common light, unless they be polarized oppositely, and have their intensities equal.
19. We have seen that the nature of the mixture of a given group of independent polarized streams is determined by the values of the four constants A, B, C, D. Consider now the mixture of a stream of common light having an intensity J, and a stream, independent of the former, consisting of elliptically polarized light having an intensity J', and having a for the azimuth of its plane of maximum polarization, and tan /3' for the ratio of the axes of the ellipse which characterizes it.
By the preceding article, the stream of common light is equivalent to two independent streams, plane-polarized in azimuths 0 and 90, having each an intensity equal to \J. Hence, applying the formulae (16) and (17) to the mixture, we have
2m (cj2 = J+J'+J' sin 2/3' sin 2ft + J cos 2a' cos 2/3' cos 2^ cos 2ft
+ J' sin 2a' cos 2/3' sin 2ax cos 2ft ;
and this mixture will be equivalent to the original group of polarized streams, provided
A;                     /'sin 2/3' = 5;      j 'cos 2/3'= 0;       /'sin 2a'cos 2/3' = D. J.....